Dynamics of a rational system of difference equations in the plane

We consider a rational system of first order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions and the stability of equilibria.


Introduction
In recent years, rational difference equations have attracted the attention of many researchers for varied reasons. On the one hand, they provide examples of non-linear equations which are, in some cases, treatable but whose dynamics present some new features with respect to the linear case. On the other hand, rational equations frequently appear in some biological models and, hence, their study is of interest also due to their applications. A good example of both facts are Ricatti difference equations; the richness of the dynamics of Ricatti equations is very well-known (see, for instance, [7] and [8]) and a particular case of these equations provides the classical Beverton-Holt model on the dynamics of exploited fish populations [3]. Obviously, higher order rational difference equations and systems of rational equations have also been widely studied but still have many aspects to be investigated. The reader can find in the following books [1], [5], [11], and the works cited therein many results, applications and open problems on higher order equations and rational systems.
A preliminar study of planar rational systems in the large can be found in the paper [6] by Camouzis et al. In such work, they give some results and provide some open questions for systems of equations of the type x n+1 = α 1 + β 1 x n + γ 1 y n A 1 + B 1 x n + C 1 y n y n+1 = α 2 + β 2 x n + γ 2 y n A 2 + B 2 x n + C 2 y n        , n = 0, 1, . . .
where the parameters are taken to be non-negative. As shown in the cited paper, some of those systems can be reduced to some Ricatti equations or to some previously studied secondorder rational equations. Further, since for some choices of the parameters one obtains a system which is equivalent to the case with some other parameters, Camouzis et al. arrived at a list of 325 non-equivalent systems to which it should be focused the attention. They list such systems as pairs (k, l) where k and l make reference to the number of the corresponding equation in their Tables 3 and 4.
In this paper, we deal with the rational system labelled as (21,23) in [6]. Note that for non-negative coefficients such system is neither cooperative nor competitive but it has the particularity that denominators in both equations are equal. This allows us to use some of the techniques developed in [2] to completely obtain the solutions and give a nice description of the dynamics of the system. In principle, we will not restrict ourselves to the case of non-negative parameters, although this case will be considered in detail in the last section. Hence, we will study the general case of the system where the parameters α 1 , α 2 , β 1 , β 2 are given real numbers and the initial condition (x 0 , y 0 ) is an arbitrary vector of R 2 . It should be noticed that when α 1 β 2 = α 2 β 1 the system can be reduced to a Ricatti equation (or it does not admit any complete solution, which occurs for α 2 = β 2 = 0) and therefore these cases will be neglected. Since we will not assume non-negativeness for neither the coefficients nor the initial conditions, a forbidden set will appear. We will give an explicit characterization of the forbidden set in each case. Obviously, all the results concerning solutions that we will state in the paper are to be apply only to complete orbits. We will focus our attention in three aspects of the dynamics of the system: the boundedness character and asymptotic behavior of its solutions, the existence of periodic orbits (and, in particular, of prime period-two solutions) and the stability of the equilibrium points. It should be remarked that, depending on the parameters, they may appear asymptotically stable fixed points, stable but not asymptotically stable fixed points, non-attracting unstable fixed points and attracting unstable fixed points. The paper is organized, besides this introduction, in three sections. Section 2 is devoted to some preliminaries and some results which can be mainly deduced from the general situation studied in [2]. Next, we study the case β 2 = 0 since such assumption yields the uncoupled globally 2-periodic equation y n+1 = α 2 /y n and the system is reduced to a linear first order equation with 2-periodic coefficients; this will be our section 3 below. The main section of the paper is section 4, where we give the solutions to the system and the description of the dynamics in the general case β 2 = 0. We finish the paper by describing the dynamics in the particular case where the coefficients and the initial conditions are taken to be non-negative.

Preliminaries and first results
Systems of linear fractional difference equations X n+1 = F (X n ) in which denominators are common for all the components of F have been studied in [2]. If one denotes by q the mapping given by q(a 1 , a 2 , . . . , a k+1 ) = (a 1 /a k+1 , a 2 /a k+1 , . . . , a k /a k+1 ) for (a 1 , a 2 , . . . , a k+1 ) ∈ R k+1 with a k+1 = 0 and ℓ : R k → R k+1 is given by ℓ(a 1 , a 2 , . . . , a k ) = (a 1 , a 2 , . . . , a k , 1), it is shown in such work that the system can be written in the form X n+1 = q • A • ℓ(X n ), where A is a (k + 1) × (k + 1) square matrix constructed with the coefficients of the system. In the special case of our system (1) one actually has This form of the system let us completely determine its solutions in terms of the powers of the associated matrix Actually, the explicit solution to the system with initial condition (x 0 , y 0 ) is given by where M t stands for the transposed of a matrix M . Therefore, our system can be completely solved and the solution starting at (x 0 , y 0 ) is just the projection by q of the solution of the linear system X n+1 = AX n with initial condition X 0 = (x 0 , y 0 , 1) t whenever such projection exists.
Remark 1 When such projection does not exist, then (x 0 , y 0 ) lies in the forbidden set. Clearly, this may only happen when for some n ≥ 1 one has (0, 0, 1)A n (x 0 , y 0 , 1) t = 0.
Therefore, if a i (n) ∈ R, 0 ≤ i ≤ 2 are such that A n = a 0 (n)I + a 1 (n)A + a 2 (n)A 2 , then one obtains immediately that the forbidden set is given by the following union of lines The explicit calculation of a i (n), 0 ≤ i ≤ 2 for each n ≥ 3 may be done in several ways. For instance, one has that a 0 (n) + a 1 (n)x + a 2 (n)x 2 is the remainder of the division of x n by the characteristic polynomial of A. Further, by elementary techniques of Linear Algebra one can also compute them in terms of the eigenvalues of A (an approach using the solutions to an associated linear difference equation may be seen in [9]).
Remark 2 As mentioned in the introduction, all through the paper we will consider that (this is to say, that the matrix A is non-singular) since the cases with β 2 α 1 = β 1 α 2 may be reduced to a single Ricatti equation. Actually, if α 2 = β 2 = 0, then the system does not admit any complete solution, whereas, for α 2 = 0 or β 2 = 0, one has that there exists a constant C such that α 1 = Cα 2 and β 1 = Cβ 2 and hence the first equation of the system may be substituted by x n+1 = Cy n+1 and then the second one reduces to the Ricatti equation Our main goal will be to give a description of the dynamics of the system in terms of the eigenvalues of the associated matrix A given in (2). We begin with the following result concerning 2-periodic solutions which is the particularization to our system of the analogous general result given in Theorem 3.1 and Remark 3.1 of [2].
3. There exist periodic solutions of prime period 2 if and only if α 1 β 2 = 0.
Proof. As stated in [2], a point (a, b) ∈ R 2 is an equilibrium if and only if (a, b, 1) is an eigenvector of the associated matrix A. When β 2 = 0, it is straightforward to prove that for each real eigenvalue λ the vector λ 2 −α2 β2 , λ, 1 is an eigenvector. In the case β 2 = 0, the equilibrium points can be easily computed directly from the equations α 2 = y 2 , For the proof of affirmation (3) it suffices to bear in mind that, according to [2], the existence of prime period-two solutions is only possible when the associated matrix A has an eigenvalue λ such that −λ is also an eigenvalue. Since A is a 3 × 3 square matrix, this obviously implies that the trace of A is also an eigenvalue. Hence, β 1 is an eigenvalue but this is only possible if α 1 β 2 = 0. If α 1 = 0, then the initial condition (0, y 0 ) gives a prime period 2 solution whenever y 2 0 = α 2 , whereas, if α 1 = 0 and β 2 = 0, a direct calculation shows that the solution with initial conditions (0, −β 1 ) is periodic of prime period 2.
We now study the stability of fixed points in some of the cases. Recall that a fixed point of our system (x * , y * ) always verifies y * = λ for some real eigenvalue λ of the matriz A. We will say in such case that the fixed point (x * , y * ) is associated to λ.
Proposition 4 Let us consider the system (1) with α 1 β 2 = α 2 β 1 . Let ρ(A) be the spectral radius of the matrix A given in (2) and let λ be an eigenvalue of A.
2. If |λ| = ρ(A) and all the eigenvalues of A whose modulus is ρ(A) are simple, then the associated fixed point is stable. Further, if in this case λ is the unique eigenvalue whose modulus is ρ(A), then it is assymptotically stable.
Proof. The Jacobian matrix of the map F (x, y) = α1+β1x y , α2+β2x y at a fixed point (x * , y * ) is given by Consider an eigenvalue λ of A and let λ 2 , λ 3 be the other (non-necessarily different) eigenvalues of A. Let us show that the eigenvalues of the Jacobian matrix at a fixed point associated to λ are just λ2 λ and λ3 λ . The result is trivial when β 2 = 0 since the eigenvalues of A are β 1 and ± √ α 2 and fixed points are always associated to one of the eigenvalues ± √ α 2 . If β 2 = 0, then x * = λ 2 −α2 β2 and y * = λ and, therefore, one obtains: showing that the eigenvalues of DF (x * , y * ) are as claimed. Now, the first statement follows at once since, if |λ| < ρ(A), then at least one of the eigenvalues of DF (x * , y * ) lies outside the unit circle. Moreover, when |λ| = ρ(A) and it is the unique eigenvalue with such property, then the eigenvalues of DF (x * , y * ) are inside the (open) unit ball and hence the equilibrium (x * , y * ) is assymptotically stable, which proves the second part of (2). For the proof of the first part of (2) let us recall that if (x * , y * ) is a fixed point of (1) associated to the real eigenvalue λ, then X * = (x * , y * , 1) t is a fixed point of the linear system X n+1 = 1 λ AX n . The eigenvalues of the matrix M = 1 λ A are obviously 1, λ2 λ and λ3 λ . Since the eigenvalues of A having modulus ρ(A) are simple, so are the eigenvalues of M having modulus 1. Therefore, the fixed point X * is stable [8,Th. 4.13]. Now, the stability of (x * , y * ) follows at once from (3) and the continuity of q in the semi-space z > 0.
Recall that, since we are assuming that inequality (4) holds, we have β 1 α 2 = 0. In this case, the forbidden set of the system reduces to the line y = 0. Since β 2 = 0 the second equation of the system becomes the uncoupled equation which, as far as α 2 = 0, for each initial condition y 0 = 0 gives Substituting such values in the first equation of the system we obtain a first order linear difference equation with 2-periodic coefficients whose solution is given by x 1 = (α 1 +β 1 x 0 )/y 0 and, for n > 1, for even n, Hence, we have proved the following.
From the proposition above one can easily derive the following result which completely describes the asymptotic behaviour of the solutions to the system. Corollary 6 Let us consider β 2 = 0 and β 1 α 2 = 0.
then any other solution of (1) is bounded and tends to one of the period-two solutions described above.
Proof. The proof is a straightforward consequence of the explicit formulas for x n and y n given in Proposition 5. It should, however, be mentioned that the globally periodicity of the system in the case β 1 2 = −α 2 can be easily seen since the associated matrix A given by (2) in such case verifies A 4 = β 4 1 I, where I stands for the identity matrix. Actually, a simple calculation proves that the solution starting at (x 0 , y 0 ) is the 4-cycle . From the above result and Proposition 4 one easily gets the following information about the stability of the fixed points.
Proposition 8 Suppose β 2 = 0 and (x 0 , y 0 ) is an initial condition not belonging to the forbidden set F. In such case the solution of system (1) is given by where v n is the unique solution of the linear difference equation with initial conditions v −1 = 1, v 0 = y 0 and v 1 = β 2 x 0 + α 2 .
Proof. As we have seen in section 2, the solution to system (1) starting at a point (x 0 , y 0 ) not belonging to the forbidden set is just the projection by q of the solution of the linear system (u n+1 , v n+1 , w n+1 ) t = A(u n , v n , w n ) t with initial condition (x 0 , y 0 , 1) t where A is given by (2). Since the third equation of such linear systems reads w n+1 = v n , it can be reduced to the planar linear system of second order equations and hence, if (u n , v n ) is the solution to (9) obtained for the initial conditions (u 0 , v 0 , v −1 ) = (x 0 , y 0 , 1), then the solution of our rational system for the initial values (x 0 , y 0 ) will be It is clear that for β 2 = 0 we have that u n can be completely determined by (9) in terms of v n+1 and v n−1 and hence it suffices to solve the third order linear equation trivially deduced from (9) and substitute the corresponding values in (10) to obtain the result claimed.
In the following results we shall discuss the behavior of the solutions to (1) by using Proposition 8. We shall consider three different cases depending on the roots of the characteristic polinomial of the linear equation (8). Recall that such roots are also the (possibly complex) eigenvalues of the matrix A given in (2).
From Proposition 8 we see that the asymptotic behavior of the solutions of system (1) will depend on the asymptotic behavior of the sequences vn vn−1 , being v n solutions of the linear difference equation (8). The Theorem of Poincaré (Theorem 8.9 in [8]) establishes a general result for the existence of lim n→∞ vn vn−1 . In our case, since equation (8) has constant coefficients, we can directly do the calculations, even in the cases not covered by the Theorem of Poincaré, to describe the dynamics of system (1).

The characteristic polinomial has no distinct roots with the same module
Let λ 1 , λ 2 , λ 3 be the three roots of the characteristic polynomial of the linear difference equation (8) in this case. A condition on the coefficients for this case can be given by: with α 1 = 0 or α 2 ≤ 0. Recall that we assume here that β 2 α 1 = β 1 α 2 and β 2 = 0. If λ 1 is the characteristic root of maximal modulus, we will denote by L the line Proposition 9 Suppose that β 2 = 0 and every root of the characteristic polinomial of the linear difference equation (8) is real and no two distinct roots have the same module. When (x 0 , y 0 ) is not in the forbidden set, we have: (c) the corresponding solution to the system with initial condition (x 0 , y 0 ) = λ 3 2 − α 2 β 2 , λ 3 and (x 0 , y 0 ) ∈ L converges to λ 2 2 − α 2 β 2 , λ 2 .
Proof. In all the cases, the equilibrium points are directly given by Proposition 3. The assertions concerning the asymptotic behaviour can be derived as a consequence of Case 1 in [8, pg. 240], bearing in mind that and that v n is the solution to the linear equation (8) with initial conditions v −1 = 1, v 0 = y 0 and v 1 = β 2 x 0 + α 2 .

The characteristic polinomial has two distinct real roots with the same module
It is easy to check that this case occurs when β 1 = 0, β 2 = 0, α 1 = 0 and α 2 > 0. Thus, the roots of the characteristic polynomial of the linear difference equation (8) are β 1 and ± √ α 2 .
Proposition 10 Suppose β 1 = 0, β 2 = 0, α 1 = 0 and α 2 > 0. Assume also that (x 0 , y 0 ) is not in the forbidden set. Proof. In all cases the affirmation a) is a consequence of Proposition 3. When β 1 2 = α 2 , the roots are β 1 , with algebraic multiplicity two, and −β 1 . By Proposition 8 we know that any solution of the system can be written as where P 1 , P 2 and P 3 actually satisfy If P 1 = 0, then (x n , y n ) obviously tends to (0, β 1 ). From (12) we see that P 1 = 0 if and only if x 0 = 0 and, in such case, x n = 0 and y n takes alternatively the values Aβ 1 and P2−P3 . Notice that y 0 = 0 guaranties P 2 + P 3 = 0 and, since β 1 = 0, we can not have P 1 = 0 and P 2 − P 3 = 0. This completes the proof of (1).
In the case β 2 1 = α 2 , by Proposition 8 we can write the general solution of the system as where P 1 , P 2 and P 3 satisfy When β 2 1 > α 2 one immediately gets the results of statement (2) with an argument similar to that of the previous case. Therefore we will focus our attention in the case β 2 1 < α 2 . The condition x 0 = 0 is, according to (13), equivalent to P 1 = 0 and in such case one gets x n = 0 and y n takes alternatively the values K √ α 2 and K −1 √ α 2 with K = P2+P3 P2−P3 = y0 α2 . Now, if P 1 = 0 and the initial conditions are taken such that P 2 + P 3 = 0 = P 2 − P 3 , then (x n , y n ) tends obviously to the 2-cycle On the contrary, if either P 2 +P 3 = 0 or P 2 −P 3 = 0 (and only one of both equalities holds) then both sequences x n and y n are unbounded. From the system (13) one gets that P 2 − P 3 = 0 if and only if x 0 = (β 2 1 − α 2 )/β 2 and that P 2 + P 3 = 0 is equivalent to This shows the validity of c).

The characteristic polinomial has complex roots
Now we consider the case in which the characteristic polynomial of the linear difference equation has a couple of complex roots ρe ±iθ , with sin θ > 0. Let λ = 0 be the real root. It can be easily shown that and that this situation occurs when By Proposition 3 we know that the unique equilibrium is Notice that (β 1 − λ)(y + λ) = 2yρ cos θ − α 2 − ρ 2 . Also observe that the equilibrium does not belong to L.
Theorem 11 Suppose β 2 = 0 and the characteristic polynomial of the linear difference equation to have complex roots and assume that (x 0 , y 0 ) is not in the forbidden set.
1. The solutions starting on the line L remain on it and they are either all periodic or all unbounded.
2. If |λ| > ρ, then the unique equilibrium attracts all the solutions not starting on L.
3. If |λ| < ρ, then every non fixed bounded subsequence of a solution accumulates on L.
4. If |λ| = ρ, then every complete solution (neither starting on the fixed point nor on L) lies on a not degenerate conic, which does not contain the equilibrium.
As a consequence, λ 2 σ n+1 − 2λρσ n cos θ + ρ 2 σ n−1 = 0, and then which is equivalent to Using (15) one has that (x 0 , y 0 ) ∈ L if and only if P = 0 and from (18) we then get that (x n , y n ) ∈ L for all n ≥ 1. Furthermore, by (16), we see that if (x 0 , y 0 ) ∈ L, then the solution (x n , y n ) is periodic whenever θ/π is a rational number and unbounded otherwise.
Assume now that the solution (x n , y n ) does not start on L, this to say, P = 0. We will now distinguish the three cases: |λ| > ρ, |λ| < ρ and |λ| = ρ.
If |λ| > ρ, then by (16) one immediately has x n → λ 2 −α2 β2 and y n → λ. Suppose now that |λ| < ρ. If (x n k , y n k ) is a subsequence satisfying that inf k |cos (a + (n k − 1) θ)| > 0, then one obviously has σ n k −1 → ∞. Using the definition of σ n , one easily gets that σ n k σ n k −1 is bounded. Then, (x n k , y n k ) is a bounded subsequence and equation (18) shows that it is attracted by the line L.
It should also be noticed that the fixed point lays on the line 10x + 2y + 3 = 0. This is also the case in the general setting. It follows from (18) that whenever σ n k −1 = 0 then the point (x n−k , y n−k ) is on the line containing the fixed point which is parallel to L.
Remark 13 Notice that, according to the results in [2], when |λ| = ρ and the argument θ of the complex root is a rational multiple of π, the system is globally periodic.

Stability of fixed points
We finish this section with the complete study of the stability of the fixed points in the case β 2 = 0.
Theorem 14 Suppose that β 2 = 0, let λ be a real eigenvalue of the matrix A given in (2). Let λ 2 −α2 β2 , λ be the associated fixed point and denote by ρ(A) the spectral radius of A.
2. If |λ| = ρ(A), then the associated equilibrium is stable if and only if every eigenvalue whose modulus is ρ(A) is a simple eigenvalue. Moreover, the stability is asymptotic if and only if λ is a simple eigenvalue and it is the unique eigenvalue of A whose modulus is ρ(A).
Proof. The first statement was already proved in Proposition 4. Besides, in such Proposition, we have shown that if every eigenvalue whose modulus is ρ(A) is simple then the associated equilibrium is stable. Let us prove the converse. According to the results of the previous subsections, the only cases in which one has a non-simple eigenvalue of maximal modulus are the cases treated in Proposition 9 (1) and (4) and the first case of Proposition 10. We will see that in such cases the equilibrium points associated to eigenvalues of maximal modulus are unstable.
Finally, it only remains to prove that when A has distinct simple eigenvalues whose modulus equal ρ(A) then the fixed point is not asymptotically stable. But this situation can only happen if either one has the situation described in Proposition 11 (4) or the one given in Proposition 10 (3). In the case of complex eigenvalues we had seen that all the orbits lie on conics not going through the fixed point and, hence, it cannot be asymptotically stable. In the other case, it is clear that the fixed points (0, ± √ α 2 ) are not attracting since every solution starting on the line x = 0 is 2-periodic.
Remark 15 It is interesting to notice that in the three cases in which there is an eigenvalue of maximal modulus with multiplicity larger than 1, the corresponding fixed point is attracting but unstable.

Non-negative solutions to the system with non-negative coefficients
When the coefficients of our system (1) are non-negative and we restrict ourselves to nonnegative initial conditions, many of the cases studied in the previous sections cannot appear.
Further, in such case one may describe which kind of orbits appear and their assymptotic behaviour without the previous calculation of the characteristic roots. It shoud be noticed that whenever the coefficients in system (1) are non-negative and α 1 β 2 = α 2 β 1 , every initial condition (x 0 , y 0 ) with x 0 ≥ 0, y 0 > 0 gives rise to a complete orbit except for α 2 = 0 where the condition x 0 > 0 is also necessary.
It will be convenient to study independently the case α 1 β 2 = 0. Next result is a simple summary of the results in Section 3 and Proposition 10 and, hence, we omit its proof.
The remaining cases are jointly treated in the following result. All the definitions and results on non-negative matrices which are used in its proof may be found in [10,Ch. 8] Proposition 17 Suppose that system (1) has non-negative coefficients and that α 1 β 2 = 0.
Proof. Let us consider A as in (2). A simple calculation shows that (A + I) 2 is positive and, therefore, A is irreducible. Then the spectral radius ρ(A) is a strictly positive simple eigenvalue of A.
If there exists another eigenvalue λ such that |λ| = ρ(A) then, since A is non-negative and irreducible, the eigenvalues of A should be λ k+1 = ρ(A)e ikπ/3 where k = 0, 1, 2 and, consequently, A 3 = ρ(A) 3 I. The direct computation of A 3 shows that this is possible if and only if α 2 = β 1 = 0 and, hence, in that case, the system is 3-periodic and the only equilibrium is the one associated to the real eigenvalue ρ(A).
In the remaining cases, λ 1 = ρ(A) is a dominant eigenvalue and, according to our results of Proposition 4, Proposition 9 and Theorem 11, the corresponding fixed point is stable and attracts all complete solutions except those starting on the line L = { (x, y) : β 2 x = (β 1 − λ 1 )(y + λ 1 ) }.